Geometric blow-up of a dynamic Turing instability in the Swift-Hohenberg equation

被引：0
作者：
Hummel, F. [1 ]
Jelbart, S. [2 ]
Kuehn, C. [1 ]
机构：
[1] Tech Univ Munich, Sch Computat Informat & Technol, D-85748 Munich, Germany
[2] Univ Adelaide, Sch Comp & Math Sci, Adelaide, SA 5000, Australia
来源：
JOURNAL OF DIFFERENTIAL EQUATIONS | 2025年 / 427卷
基金：
欧盟地平线“2020”;
关键词：
Swift-Hohenberg equation; Geometric blow-up; Modulation theory; Singular perturbation theory; Ginzburg-Landau equation; Delayed stability loss; GINZBURG-LANDAU EQUATION; PERTURBED PARABOLIC EQUATION; MODULATION EQUATIONS; PERIODIC-SOLUTIONS; UNBOUNDED-DOMAINS; GLOBAL EXISTENCE; SLOW MANIFOLDS; STABILITY LOSS; BIFURCATION; VALIDITY;
D O I：
10.1016/j.jde.2025.01.036
中图分类号：
O1 [数学];
学科分类号：
0701 ; 070101 ;
摘要：
We consider the slow passage through a Turing bifurcation in the Swift-Hohenberg equation. We generalise the formally derived multiple scales ansatz from modulation theory for use in the slowly time- dependent setting. The key technique is to reformulate the problem via a geometric blow-up transformation. This leads to non-autonomous modulation equations of Ginzburg-Landau type in the blown-up space. We analyse solutions to the modulation equations in weighted Sobolev spaces in two different cases: (i) A symmetric case featuring delayed stability loss, and (ii) A non-symmetric case with a source term. Rigorous estimates on the error of the dynamic modulation approximation are derived in order to characterise the dynamics of the Swift-Hohenberg equation. This allows for a detailed asymptotic description of solutions to the original Swift-Hohenberg equation in both cases (i)-(ii). We also prove the existence of delayed stability loss in case (i), and provide a lower bound for the delay time.<br /> (c) 2025 Published by Elsevier Inc.
引用
收藏
页码：219 / 309
页数：91
相关论文
共 99 条
  • [21] Chossat P., Iooss G., The Couette-Taylor Problem, vol. 102, (2012)
  • [22] Collet P., Eckmann J.-P., The time dependent amplitude equation for the Swift-Hohenberg problem, Commun. Math. Phys., 132, pp. 139-153, (1990)
  • [23] Collet P., Eckmann J.-P., Instabilities and fronts in extended systems, Instabilities and Fronts in Extended Systems, (1990)
  • [24] Cross M.C., Hohenberg P.C., Pattern formation outside of equilibrium, Rev. Mod. Phys., 65, (1993)
  • [25] De Maesschalck P., Kaper T.J., Popovic N., Canards and bifurcation delays of spatially homogeneous and inhomogeneous types in reaction-diffusion equations, Adv. Differ. Equ., 14, pp. 943-962, (2009)
  • [26] Dean A.D., Matthews P., Cox S., King J., Exponential asymptotics of homoclinic snaking, Nonlinearity, 24, (2011)
  • [27] DiPrima R., Eckhaus W., Segel L., Non-linear wave-number interaction in near-critical two-dimensional flows, J. Fluid Mech., 49, pp. 705-744, (1971)
  • [28] Dumortier F., Roussarie R., Canard Cycles and Center Manifolds, Memoirs of the American Mathematical Society, 577, (1996)
  • [29] Eckhaus W., The Ginzburg-Landau manifold is an attractor, J. Nonlinear Sci., 3, pp. 329-348, (1993)
  • [30] Eckmann J.-P., Schneider G., Non-linear stability of modulated fronts for the Swift–Hohenberg equation, Commun. Math. Phys., 225, pp. 361-397, (2002)
12345678910